Observe that, with \(a\)  (resp. \(b\) ) denoting the singleton trace whose
only event is labelled \(a\)  (resp. \(b\) ), if \((a,b) \in I\)  then \(ab=ba\)  in \(TR(\widetilde{\Sigma })\) .
A basic result in trace theory gives a presentation of the trace monoid as a quotient of
the free word monoid \(\Sigma ^*\)  by the congruence \({\sim _I} \subseteq \Sigma ^*\times \Sigma ^*\)  generated by \(ab \sim _I ba\)  for \((a,b) \in I\) .
See [1]} for more details.
